Computer Aided Engineering Design
DESIGN OF CURVES 103 The computation of intermediate de Casteljau’s points for degree n Bézier curve can be illustrated by the t ...
104 COMPUTER AIDED ENGINEERING DESIGN (b) Partition of Unity and Barycentric Coordinates: Irrespective of the values of u, the B ...
DESIGN OF CURVES 105 (c) Symmetry: Bu Bin( ) = nin–(1 – )u (4.36) Though suggested in Figure 4.14, the property is shown as foll ...
106 COMPUTER AIDED ENGINEERING DESIGN = ( – 1)! ( – 1)!( – )! (1 – ) – ( – 1)! !( – 1 – )! n – –1 (1 – )–1– n ini uu n in i ⎡ ni ...
DESIGN OF CURVES 107 ′≡ ′ ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ′≡ ′ ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ A ⎥ x y xp yq B x y xp yq = and = 1 ...
108 COMPUTER AIDED ENGINEERING DESIGN For a composite curve, individual segments need to be of lower order, preferably cubic. Th ...
DESIGN OF CURVES 109 The transformed Bézier segment is plotted in Figure 4.16 (b). Observe that shape of the segment does not ch ...
110 COMPUTER AIDED ENGINEERING DESIGN ̇rP() = () = P[ () – ()] =0 =0 –1 u d –1 –1 du Bu nB u B u j n j j n j n j j n j ΣΣn = =0 ...
DESIGN OF CURVES 111 (e) Variation Diminishing: For a planar Bézier segment, it can be verified geometrically that no straight l ...
112 COMPUTER AIDED ENGINEERING DESIGN Thus, the derivative of a Bézier segment is a degree n–1 Bézier segment with control point ...
DESIGN OF CURVES 113 DD D P P^2 i = i^11 +1 – i = ( ii i+ 2 – +1) – (P P P P P+1 – ) = i ii+ 2 – 2 +1 + i DD D P^3 i = i^22 +1 – ...
114 COMPUTER AIDED ENGINEERING DESIGN As an example, the control points for a cubic Bézier segment in 0 ≤u≤c are determined. Fro ...
DESIGN OF CURVES 115 k = 0: DQnn^00 = ⇒qn = bn k = 1: (1 – c)DQnn^1 –1 =^1 –1 ⇒ (1 – c) (bn – bn–1) = (qn – qn–1) ⇒qn–1 = (1 – c ...
116 COMPUTER AIDED ENGINEERING DESIGN rQr R( ) = 1 ( ), ( ) = ( ) =1 3 3 (^122) =1 3 3 uBuuBuiΣΣi i i i i 2 Defining the paramet ...
DESIGN OF CURVES 117 for Bézier curve of degree n defined by control points b 0 ,b 1 ,... , bn, to raise its degree by one requi ...
118 COMPUTER AIDED ENGINEERING DESIGN converted to the cubic Bézier form and vice-versa. Given control points Pi,i = 0,... , 3, ...
DESIGN OF CURVES 119 Eq. (4.58) implies that with the first segment given, the position continuity constraints the first control ...
120 COMPUTER AIDED ENGINEERING DESIGN Using position and slope continuity conditions in Eq. (4.60) yields tr (1) = tr (0) 2 1 2 ...
DESIGN OF CURVES 121 four data points can be chosen freely. For the second segment, three of the four points, namely q 0 ,q 1 an ...
122 COMPUTER AIDED ENGINEERING DESIGN PPH i n i n i H i n i n ii ii ii i t Xt Yt Zt Wt Bt Bt wx wy wz w () () () () () = ( ) = ( ...
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